You asked for it and here it is.
So let me explain what this is. I did a Batting Game Finder search for all games with HR=1 as well as all games HR=2. That refers to individual player performances, not team totals. So, any one game could register 5, 6, 7 or more times if that many players hit 1 or 2 HR in the game. Total team performances, as well as any games where an individual hit 3 or more HR, are ignored. (Of course, 3+ HR games are extremely rare by comparison to 1 and 2 HR games.)
Then, for each year, I divided 1 HR games by 2 HR games. As you can see, in 1956 there were approximately 15 times as many 1 HR games as 2 HR games. Through about 1985, the ratio generally increased. It's very difficult to tell whether that was due to generally more 1 HR games, or generally fewer 2 HR games, since the HR totals overall vary from year to year.
I note that 1987 shows a local valley for ratio. In this year, balls were flying out and a lot of players who didn't often hit homers hit them out this year. Lots of guys had seasons with 10 or 20 homers who rarely hit more than 5 in a year. At those total rates, it's likely that most of the HR came one game at a time, and therefore the number of 1 HR games was likely up significantly while the number of 2 HR games was probably up just a bit.
And then what do we see in 1993? This is undoubtedly the beginning of The Steroid Era, as much data and analysis has shown. We see not only did the ratio drop off, but the noise from year to year has been much smaller than for the 40 or so years prior. I think this is easily explainable: many more HRs have been hit, with lots of guys amassing 40+ in a year. With these totals, the likelihood of 2 HR games is much higher than with just 10 or 20 HRs in a season, and therefore more 2 HR games are happening, dropping the ratio a bit. (Mind you, 1 HR games are still about 15 times more common.) The other reason for the drop in year-to-year noise is because more HRs are being hit. We have more events spread over the same 162-game schedule, so things tend to even out. For those not statistically inclined, just think of the opposite--counting a really rare event, such as a no-hitter. Some years there are none, and some years there are 5. The year-to-year noise is huge. But for much more common events, the rates tend to even out and be more regular from year to year.
So there you have it.